Model parameterization as a building block for image-based predictions of mechanical properties

As high resolution 3D images are getting increasingly incorporated into core analysis workflows, strategies for improving the connection between parameters in effective medium models and accessible metrics in image volumes are becoming more and more pertinent. The area of rock strength modeling in particular is rich in options for failure and yield criteria, but typically lacks parameters that are constrained by microstructural observables. Here, we apply a recently proposed strength model to a collection of published data sets and offer some suggestions as to how the microstructural control over the model parameters could be elucidated using X-ray CT images.

The strength model used in this article was presented in a previous publication. It allows to delineate strength envelopes both in the brittle and compactant domains with just three parameters. The formulation honors the fundamental difference between the kinematics of shear and compactive yielding with two different laws that employ the same stress sensitivity function. The three parameters of the model are the unconfined compression strength (UCS), the stress sensitivity factor xi and the compactive to shear strength ratio at the intergranular contacts gamma.

The stress sensitivity factor xi is responsible for the increase in effective contacting surface within the aggregate with hydrostatic pressure. It acts as a scaling factor on the strength envelopes and is also responsible for their non linearity. The parameter gamma effectively controls the location at which the strength envelope transitions from brittle to compactive yield, i.e. the stress level at which it becomes easier to fail in compaction rather than by macroscopic shear. The figure above provides an illustration of the strength envelopes obtained by the model for various combinations of xi and gamma values, with a constraint on P*, the value of the pressure for the onset of grain crushing and pore collapse under hydrostatic loading.

The model can be applied to existing data sets in view of extracting xi and gamma (as well as an apparent anisotropy factor if desired) either for rock typing or as a starting point for investigating the connection with the microstructures. The figure showed below provides the values extracted from published strength data obtained in sandstone. On the left, the stress sensitivity factor for all data sets is plotted against the porosity, which emerges as a first order control. On the right, we simply reports the extent of the gamma values range with a cartoon showing the corresponding change in the shape of the strength envelope.

Although xi and gamma could certainly be parameterized further using metrics found in existing effective medium models (granular and/or inclusion-based), we seek to evaluate how many real observables can be quantified based on high resolution X-ray CT images. Aside from the intrinsic properties of an aggregate's components, the nature of the mechanical contacts achieved among those components and their evolution with stress and strain are of utmost importance for effective properties prediction. We propose to apply the concept of rock structure fingerprinting to the solid portion of a segmented high resolution X-ray CT image volume as a step towards diagnosing the state of these solid contacts within the aggregate.

The concept of rock structure fingerprinting based on 3D images was introduced in a previous article in the form of a 2D histogram which combines the result of a digital mercury intrusion porosimetry (MIP) experiment with the direct distribution of the pore space local thickness values. This visualization helps address the ambiguity associated with incremental injection steps in routine MIP data (i.e. whether some of the injected volume corresponds to the invasion of large pore bodies compared to the access radius at a given step), while providing a unique snapshot of the pore space architecture.

When applied to the solid fraction of a rock image, the same analysis can provide critical information regarding the intergranular contacts, their connectivity and their relationship to grain size. The figure above can be considered as the companion figure to the one showed in our previous article as it was obtained from the same 3D image of Castlegate sandstone. Similarly to natural pore networks, the aggregate itself possesses a characteristic contact radius that globally represents the contact size through which the grains are interconnected. The figure also shows the importance of the shielded grain fraction in achieving through-sample connectivity, what should matter greatly to strength estimates. Though much work remains to be done to close the loop and link these diagrams to the xi and gamma parameters, there is great promise in this approach in the sense that it offers a statistical snapshot of the rock structure, which could be used as a canvas to run deformation scenarios.

Our objective here was to emphasize the importance of combining meaningful attributes in experimental data with what can actually be quantified in image volumes for the successful design of image-based mechanical models. The description of the rock structure that we offered is essentially static, but many techniques are available today to spatially monitor the deformation process and must be taken advantage of in coming up with new measurables, and hence new models, which should explicitly address the role of spatial heterogeneity.

Thank you for reading us, happy holidays, and all the best in the new year!

Laurent Louis, Adage Corporation, 23 December 2018.